Final answer:
The error in writing a polynomial function with rational coefficients and the given zero 2i is that it should include the conjugate of 2i, which is -2i.
Step-by-step explanation:
The error in writing a polynomial function with rational coefficients and the given zero 2i lies in the fact that if 2i is a zero, then its conjugate, -2i, must also be a zero. This is because complex roots of polynomials with real coefficients always come in conjugate pairs.
If 2i is a zero of the polynomial, the corresponding factor would be (x - 2i), and if (-2i) is also a zero, the corresponding factor would be (x + 2i).
Therefore, the correct factorization considering both roots is (x - 2i)(x + 2i).
If there was an error and only (x - 2i) was considered, then the polynomial would not have real coefficients, violating the requirement specified. The correct representation would include both factors (x - 2i)(x + 2i) to ensure rational coefficients.